Closure Properties of Solutions to Heat Inequalities

Abstract

We prove that if u 1,u 2:(0,∞)×ℝ d →(0,∞) are sufficiently well-behaved solutions to certain heat inequalities on ℝ d then the function u:(0,∞)×ℝ d →(0,∞) given by $u^{1/p}=u_{1}^{1/p_{1}}*u_{2}^{1/p_{2}}$ also satisfies a heat inequality of a similar type provided $\frac{1}{p_{1}}+\frac{1}{p_{2}}=1+\frac{1}{p}$ . On iterating, this result leads to an analogous statement concerning n-fold convolutions. As a corollary, we give a direct heat-flow proof of the sharp n-fold Young convolution inequality and its reverse form.